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## G = C2×C40⋊C2order 160 = 25·5

### Direct product of C2 and C40⋊C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C2×C40⋊C2
 Chief series C1 — C5 — C10 — C20 — D20 — C2×D20 — C2×C40⋊C2
 Lower central C5 — C10 — C20 — C2×C40⋊C2
 Upper central C1 — C22 — C2×C4 — C2×C8

Generators and relations for C2×C40⋊C2
G = < a,b,c | a2=b40=c2=1, ab=ba, ac=ca, cbc=b19 >

Subgroups: 280 in 68 conjugacy classes, 33 normal (17 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×2], C22, C22 [×4], C5, C8 [×2], C2×C4, C2×C4, D4 [×3], Q8 [×3], C23, D5 [×2], C10, C10 [×2], C2×C8, SD16 [×4], C2×D4, C2×Q8, Dic5 [×2], C20 [×2], D10 [×4], C2×C10, C2×SD16, C40 [×2], Dic10 [×2], Dic10, D20 [×2], D20, C2×Dic5, C2×C20, C22×D5, C40⋊C2 [×4], C2×C40, C2×Dic10, C2×D20, C2×C40⋊C2
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, SD16 [×2], C2×D4, D10 [×3], C2×SD16, D20 [×2], C22×D5, C40⋊C2 [×2], C2×D20, C2×C40⋊C2

Smallest permutation representation of C2×C40⋊C2
On 80 points
Generators in S80
(1 77)(2 78)(3 79)(4 80)(5 41)(6 42)(7 43)(8 44)(9 45)(10 46)(11 47)(12 48)(13 49)(14 50)(15 51)(16 52)(17 53)(18 54)(19 55)(20 56)(21 57)(22 58)(23 59)(24 60)(25 61)(26 62)(27 63)(28 64)(29 65)(30 66)(31 67)(32 68)(33 69)(34 70)(35 71)(36 72)(37 73)(38 74)(39 75)(40 76)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)
(1 77)(2 56)(3 75)(4 54)(5 73)(6 52)(7 71)(8 50)(9 69)(10 48)(11 67)(12 46)(13 65)(14 44)(15 63)(16 42)(17 61)(18 80)(19 59)(20 78)(21 57)(22 76)(23 55)(24 74)(25 53)(26 72)(27 51)(28 70)(29 49)(30 68)(31 47)(32 66)(33 45)(34 64)(35 43)(36 62)(37 41)(38 60)(39 79)(40 58)

G:=sub<Sym(80)| (1,77)(2,78)(3,79)(4,80)(5,41)(6,42)(7,43)(8,44)(9,45)(10,46)(11,47)(12,48)(13,49)(14,50)(15,51)(16,52)(17,53)(18,54)(19,55)(20,56)(21,57)(22,58)(23,59)(24,60)(25,61)(26,62)(27,63)(28,64)(29,65)(30,66)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72)(37,73)(38,74)(39,75)(40,76), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (1,77)(2,56)(3,75)(4,54)(5,73)(6,52)(7,71)(8,50)(9,69)(10,48)(11,67)(12,46)(13,65)(14,44)(15,63)(16,42)(17,61)(18,80)(19,59)(20,78)(21,57)(22,76)(23,55)(24,74)(25,53)(26,72)(27,51)(28,70)(29,49)(30,68)(31,47)(32,66)(33,45)(34,64)(35,43)(36,62)(37,41)(38,60)(39,79)(40,58)>;

G:=Group( (1,77)(2,78)(3,79)(4,80)(5,41)(6,42)(7,43)(8,44)(9,45)(10,46)(11,47)(12,48)(13,49)(14,50)(15,51)(16,52)(17,53)(18,54)(19,55)(20,56)(21,57)(22,58)(23,59)(24,60)(25,61)(26,62)(27,63)(28,64)(29,65)(30,66)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72)(37,73)(38,74)(39,75)(40,76), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (1,77)(2,56)(3,75)(4,54)(5,73)(6,52)(7,71)(8,50)(9,69)(10,48)(11,67)(12,46)(13,65)(14,44)(15,63)(16,42)(17,61)(18,80)(19,59)(20,78)(21,57)(22,76)(23,55)(24,74)(25,53)(26,72)(27,51)(28,70)(29,49)(30,68)(31,47)(32,66)(33,45)(34,64)(35,43)(36,62)(37,41)(38,60)(39,79)(40,58) );

G=PermutationGroup([(1,77),(2,78),(3,79),(4,80),(5,41),(6,42),(7,43),(8,44),(9,45),(10,46),(11,47),(12,48),(13,49),(14,50),(15,51),(16,52),(17,53),(18,54),(19,55),(20,56),(21,57),(22,58),(23,59),(24,60),(25,61),(26,62),(27,63),(28,64),(29,65),(30,66),(31,67),(32,68),(33,69),(34,70),(35,71),(36,72),(37,73),(38,74),(39,75),(40,76)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)], [(1,77),(2,56),(3,75),(4,54),(5,73),(6,52),(7,71),(8,50),(9,69),(10,48),(11,67),(12,46),(13,65),(14,44),(15,63),(16,42),(17,61),(18,80),(19,59),(20,78),(21,57),(22,76),(23,55),(24,74),(25,53),(26,72),(27,51),(28,70),(29,49),(30,68),(31,47),(32,66),(33,45),(34,64),(35,43),(36,62),(37,41),(38,60),(39,79),(40,58)])

46 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 5A 5B 8A 8B 8C 8D 10A ··· 10F 20A ··· 20H 40A ··· 40P order 1 2 2 2 2 2 4 4 4 4 5 5 8 8 8 8 10 ··· 10 20 ··· 20 40 ··· 40 size 1 1 1 1 20 20 2 2 20 20 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

46 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 D4 D4 D5 SD16 D10 D10 D20 D20 C40⋊C2 kernel C2×C40⋊C2 C40⋊C2 C2×C40 C2×Dic10 C2×D20 C20 C2×C10 C2×C8 C10 C8 C2×C4 C4 C22 C2 # reps 1 4 1 1 1 1 1 2 4 4 2 4 4 16

Matrix representation of C2×C40⋊C2 in GL3(𝔽41) generated by

 40 0 0 0 1 0 0 0 1
,
 1 0 0 0 27 28 0 13 18
,
 1 0 0 0 1 34 0 0 40
G:=sub<GL(3,GF(41))| [40,0,0,0,1,0,0,0,1],[1,0,0,0,27,13,0,28,18],[1,0,0,0,1,0,0,34,40] >;

C2×C40⋊C2 in GAP, Magma, Sage, TeX

C_2\times C_{40}\rtimes C_2
% in TeX

G:=Group("C2xC40:C2");
// GroupNames label

G:=SmallGroup(160,123);
// by ID

G=gap.SmallGroup(160,123);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,218,50,579,69,4613]);
// Polycyclic

G:=Group<a,b,c|a^2=b^40=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^19>;
// generators/relations

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